Question 1202194
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The area of a rectangle is 45m^2, and the length of the rectangle is 1m more 
than twice the width. Find the dimensions of the rectangle.
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        I will show you a simple way to solve the problem  MENTALLY.



<pre>
Let W be the width, in meters; then the length is (2W+1) meters.


The area equation is

    W*(2W+1) = 45.


Multiply both sides by 2.  You will get

    2W*(2W+1) = 90.    (1)


So, the product of two numbers, (2W) and (2W+1), is 90, and the numbers differ by 1.
At this point, you can easily guess the solution: it is 2W = 9  and  W = 9/2 = 4.5.


Next, left side of equation (1) is monotonically increasing function for positive W.
It means that equation (1) has a UNIQUE solution in positive numbers, which we just found.


<U>ANSWER</U>. The problem has a unique solution: the width is 4.5 m; the length is 2*4.5+1 = 10 m.
</pre>

Solved.


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My reasoning in this solution is mathematically and logically absolutely strict,
although it includes an element of guessing; but this guessing is supported
by another reasoning, which shows that the guessed value is unique.


It is much easier than to factor a quadratic equation on the way
(which you practically will not be able to factor mentally without an error, in any case).


Such approach works for many other similar problems.



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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;About factoring, I always repeat my standard joke: 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;it works especially good, when you know an answer in advance.